Found problems: 85335
2017 Greece Junior Math Olympiad, 4
A group of $n$ people play a board game with the following rules:
1) In each round of the game exactly $3$ people play
2) The game ends after exactly $n$ rounds
3) Every pair of players has played together at least at one round
Find the largest possible value of $n$
2015 Thailand TSTST, 2
Fix a sequence $ a_1,a_2,a_3,... $ of integers satisfying the following condition:for all prime numbers $ p $ and all positive integers $ k $, we have $ a_{pk+1}=pa_k-3a_p+13 $.Determine all possible values of $ a_{2013} $.
2007 Moldova Team Selection Test, 3
Let $M, N$ be points inside the angle $\angle BAC$ usch that $\angle MAB\equiv \angle NAC$. If $M_{1}, M_{2}$ and $N_{1}, N_{2}$ are the projections of $M$ and $N$ on $AB, AC$ respectively then prove that $M, N$ and $P$ the intersection of $M_{1}N_{2}$ with $N_{1}M_{2}$ are collinear.
2006 IMO Shortlist, 6
Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.
2023 Miklós Schweitzer, 6
Prove that for all sufficiently large positive integers $n$ and a positive integer $k \leq n$, there exists a positive integer $m$ having exactly $k$ divisors in the set $\{1,2, \ldots, n\}$.
2021 Saudi Arabia JBMO TST, 3
Consider the sequence $a_1, a_2, a_3, ...$ defined by $a_1 = 9$ and
$a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}$
for $n \ge 1$.
Find all natural numbers $n$ for which $a_n$ is a perfect square of an integer.
1990 IMO Longlists, 77
Let $a, b, c \in \mathbb R$. Prove that
\[(a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2) \geq (ab + bc + ca)^3.\]
When does the equality hold?
2013 NIMO Problems, 4
Let $\mathcal F$ be the set of all $2013 \times 2013$ arrays whose entries are $0$ and $1$. A transformation $K : \mathcal F \to \mathcal F$ is defined as follows: for each entry $a_{ij}$ in an array $A \in \mathcal F$, let $S_{ij}$ denote the sum of all the entries of $A$ sharing either a row or column (or both) with $a_{ij}$. Then $a_{ij}$ is replaced by the remainder when $S_{ij}$ is divided by two.
Prove that for any $A \in \mathcal F$, $K(A) = K(K(A))$.
[i]Proposed by Aaron Lin[/i]
2011 NZMOC Camp Selection Problems, 2
Let an acute angled triangle $ABC$ be given. Prove that the circles whose diameters are $AB$ and $AC$ have a point of intersection on $BC$.
1998 Spain Mathematical Olympiad, 3
Determine the values of $n$ for which an $n\times n$ square can be tiled with pieces of the type [img]http://oi53.tinypic.com/v3pqoh.jpg[/img].
2023 VIASM Summer Challenge, Problem 4
Let $ABC$ be a non-isosceles acute triangle with $(I)$ be it's incircle. $D, E, F$ are the touchpoints of $(I)$ and $BC, CA, AB,$ respectively. $P$ is the perpendicular projection of $D$ on $EF.$ $DP$ intersects $(I)$ at the second point $K,L$ is the perpendicular projection of $A$ on $IK.$ $(LEC), (LFB) $ intersects $(I)$ the second time at $M, N,$ respectively.
Prove that $M, N, P$ are collinear.
2019 Kosovo National Mathematical Olympiad, 4
Find all real numbers $x,y,z$ such that satisfied the following equalities at same time:
$\sqrt{x^3-y}=z-1 \wedge \sqrt{y^3-z}=x-1\wedge \sqrt{z^3-x}=y-1$
2024-25 IOQM India, 21
An integer $n$ such that $\Bigl\lfloor \frac{n}{9} \Bigr\rfloor$ is a three digit number with equal digits, and $\Bigl\lfloor \frac{n-172}{4} \Bigr\rfloor$ is a $4$ digit number with the digits $2,0,2,4$ in some order. What is the remainder when $n$ is divided by $100$?
2009 Italy TST, 2
Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.
1991 Vietnam National Olympiad, 2
Let $G$ be centroid and $R$ the circunradius of a triangle $ABC$. The extensions of $GA,GB,GC$ meet the circuncircle again at $D,E,F$. Prove that:
$\frac{3}{R} \leq \frac{1}{GD} + \frac{1}{GE} + \frac{1}{GF} \leq \sqrt{3} \leq \frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}$
1935 Moscow Mathematical Olympiad, 012
The unfolding of the lateral surface of a cone is a sector of angle $120^o$. The angles at the base of a pyramid constitute an arithmetic progression with a difference of $15^o$. The pyramid is inscribed in the cone. Consider a lateral face of the pyramid with the smallest area. Find the angle $\alpha$ between the plane of this face and the base.
1988 Bulgaria National Olympiad, Problem 6
Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$.
Denmark (Mohr) - geometry, 2001.3
In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$.
[img]https://1.bp.blogspot.com/-WD8WXIE6DK4/XzcC9GYsa6I/AAAAAAAAMXg/vl2OrbAdChEYrRpemYmj6DiOrdOSqj_IgCLcBGAsYHQ/s178/2001%2BMohr%2Bp3.png[/img]
2022 HMNT, 7
All positive integers whose binary representations (excluding leading zeroes) have at least as many $1$’s as $0$’s are put in increasing order. Compute the number of digits in the binary representation of the $200$th number.
2010 Argentina National Olympiad, 1
Given several integers, the allowed operation is to replace two of them by their non-negative difference. The operation is repeated until only one number remains. If the initial numbers are $1, 2, … , 2010$, what can be the last remaining number?
2004 Federal Math Competition of S&M, 2
Let $r$ be the inradius of an acute triangle. Prove that the sum of the distances from the orthocenter to the sides of the triangle does not exceed $3r$
2022 BMT, Tie 1
For all $a$ and $b$, let $a\clubsuit b = 3a + 2b + 1$. Compute $c$ such that $(2c)\clubsuit (5\clubsuit (c + 3)) = 60$.
2014 AMC 8, 11
Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }10$
1993 AMC 8, 23
Five runners, $P$, $Q$, $R$, $S$, $T$, have a race, and $P$ beats $Q$, $P$ beats $R$, $Q$ beats $S$, and $T$ finishes after $P$ and before $Q$. Who could NOT have finished third in the race?
$\text{(A)}\ P\text{ and }Q \qquad \text{(B)}\ P\text{ and }R \qquad \text{(C)}\ P\text{ and }S \qquad \text{(D)}\ P\text{ and }T \qquad \text{(E)}\ P,S\text{ and }T$
2019 New Zealand MO, 5
Find all positive integers $n$ such that $n^4 - n^3 + 3n^2 + 5$ is a perfect square.